Question: What is the area of the region defined by the equation $x^2+y^2 - 7 = 4y-14x+3$?
Explanation: We rewrite the equation as $x^2 + 14x + y^2 - 4y = 10$ and then complete the square, resulting in  $(x+7)^2-49 + (y-2)^2-4=10$, or $(x+7)^2+(y-2)^2=63$. This is the equation of a circle with center $(-7, 2)$ and radius $\sqrt{63},$ so the area of this region is $\pi r^2 = \boxed{63\pi}$.